TH 5608 
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Copy 1 





LIBRARY OF CONGRESS. 

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©fcap- . Cqtijrigfyt ^ri + 

Shelf ».J^A r 1-2- 

UNITEI) STATES OF A'[ERICA. 







































POCKET COMPANION 


CONSISTING OF 


The most useful rules and memokan 

DAS, COLLECTED FROM SOME OF THE 
BEST ARCHITECTURAL WORKS OF 
THE DAY, AND PRACTICALLY 
TESTED BY MANY YEARS’ 
EXPERIENCE IN THE 


SHOP, FACTORY AND BUILD] NO. 


ALSO 


A TREATISE ON THE 


FRAMING SQUARE. 


BY THOMAS MOLONEY, 




CARPENTER AND JOINER. 











Entered according to Act of Congress, 
In the year 1878, by 
THOMAS MOLONEY, 

In the Office of the Librarian of Congress 
At Washington, D. C. 








\ 


INDEX. 


P.\GE 

Air, ash, atmosphere.37 

Angles, polygons &c.21, 53 

Areas &c.76 

Arithmetical signs.75 

Arcs, segments, circles, /ones, &c.30, 33 

Arch, to draw lines to or from circum’ce, 17 

Beech, brass, brick, bridges.37, 38 

Bevelled work, miters for ..51 

Bins for grain.92, 93 

Board rule, the.42 

Boxes, content of various sizes.101 

Brace rule, the.43 

Bricklayer’s memoranda.94 

Building materials, tints to express.102 

Cast iron, coal, copper.38 

Circle, the.11, 13 

“ its relative proportions.76 

“ proportions of.20 

“ radius of.17 

“ to get the length of a quarter.16 

“ to inscribe in a triangle.15 























IV 


IN DEN- 


PAGE 

Centers for striking segments.14, 73 

Chords, scale of.29 

Common rafters, to lay out.45 

“ M to get the length.46 

Compression, resistance to.66 

Content of boxes.101 

Cripple studs.50 

Cross multiplication.91 

Cross strains, resistance to.70 

Crown moulding, to cut a rake.72 

Cubic, or solid measure.101 

Curious and useful contractions.88, 91 

Dadoing.19 

Decimal fractions.81, 83 

Diagonals, scale of.43 

Diameter of pulleys.98 

Doors, right and left hand.100 

Door butts, right and left hand ..100 

Dry measure.92 

Duodecimals.83, 84 

Earth.38 

Elements of machinery.95, 97 

Ellipse, to draw an.13 

Elliptic arch, to describe.16 

Falling bodies, laws of.97 

Flaring work, miters for.52 

Fractions.86, 87 

Framing .62, 72 

Framing square, the .42, 53 































INDEX 


V 


PAGE 

Geometry, practical.9,18 

General rules, speed of pulleys.98 

Grain, bins for .92, 93 

Granite, gravel, gun metal.39 

Hints on scroll drawing.53, 58 

Hip rafters.47, 48 

Horse power.39 

Inverse proportion. 79 

Jack rafters.49 

Laws of falling bodies.97 

Lead, specific gravity, weight &c.39 

Length of circle. 15, 16 

“ “ rafters .46, 49 

Memoranda:— various materials.37 

Mensuration of solids.34, 36 

“ surfaces.25, 27 

Miters for bevelled work.51, 52 

Mouldings &c., proportions of.58 

Nails, number per pound.40, 41 

Octagons, to lay out.24, 44 

Paint, to remove old.99 

Pipes, quantity of water in.99 

Polygons, angles of &c.21, 23, 28 

Projections &c.,. ..58, 61 

Pulleys, speed and diameter of.98 

Putty, to soften old hard.99 

Quantity of water in pipes .99 

Quarter circle, length of.16 

Badius for segments.14, 73 































VI 


INDEX 


PAGE 

Radius of circles.17 

Rafters.42, 50 

Relative weights of metals.93 

Resistance of timber.65 

“ to compression.60 

“ tension*..69 

“ “ cross strains.71 

Saw-kerfing.18 

Segments.14, 73 

Scroll drawing, hints on.53, 58 

Scroll, a true taper.57 

Slate .39 

Slide rule.77, 80 

Stretchout of circle.20 

Square measure...101 

Table of angles of polygons.28 

Table of measures. 101 

Tints used by Architects in drawing.102 

Valley rafters.49 

Water, wind.39, 40 

Weight of water.94 

Wine measure.92 

Zones &c., .30, 33 


ERRATA. 

ige 12, first line, for A C B, read A T 
“ “ second line, for B F, read B T. 

“ 13, third line, for B F, read B E. 

































In ottering this little volume to the 
Building Fraternity, the Author seeks to 
supply a vacancy in Architectural litern- 
ture, viz: a cheap, and small work eontai- 
ning, in a condensed form, the most prac¬ 
tical rules to be found in many of the 
more costly books. 

The works of Nicholson, Benjamin 
Haviland,, Sloan and Hatfield have been 
freely consulted, and many valuable items 
selected from them. 

Some of the Tables and memorandas 
are taken from Voodks’ Architects’ 
Companion— the name of the Author be¬ 
ing proof of their accuracy. 

The parts relating to the Uses of the 
Protractor, Plain Scale, Slide Rule, 
Compass and Square, have been thoro¬ 
ughly tested by practical applications. 

The a plication of the Framing-Square 
in getting lengths, bevels, pitches, miters 
&c. for all square, bevelled or raking work 
has been used personally for several years 
and obviates the necessity of using as 
many lines as is often done in laying out 


PREFACE 


roofs containing Hips, Jacks, Valleys &<*. 

Many of our most valuable ‘‘Carpextei/s 
Assistant’s” are large works,— very good 
to study and commit to memory. — but 
are often of no account as a ‘Companion*, 
being either ‘at home in the Bureau’ or at 
the Shop in the Chest, when the needful 
information is wanted on an outside jol . 

To those who want a book for daily 
refference, this little volume Is offered, 
hoping that it may prove of practical 
utility to all who are interested in ti e 
Building trade 


T. M. 


THE 


CARPENTER,S AND JOINER,§ 

POCKET COMPANION. 


‘Cvnrticul ('■.ronu'ivu 


To Bisect a given Line, A B, n<;. I. 



With A and B as centers, with any 
Badius greater than one half A B, draw- 
arcs 0 E D and C F D, and through the 
points of intersection draw 0 1>. 



10 THE CARPENTER'S AND JOINER'S 




To bisect a given Angle ABO, fig. 2 


From B as a center, with any radius, 
describe arcs cutting the sides at D and E ; 
from these two points as centers, with the 
same or any other radius, draw arcs cul t, 
ing each other at F. Draw B F. 

To erect a Perpendicular at, or near the 
end of a given Line A F, fig. 3. 


From the point A, with any radius, des. 
cribe the arc BCD, From B. with the 
same radius, cross at O; and from (’. cross 
at D; then from C and 1) cross at E. 






POCKET COMPANION. 


n 


THE CIRCLE. fic t . 4. 



The boundary or margin of a Circle is 
called the circumference; the line A B 
the diameter; the line A O the radius; 
a line connecting the extremities of an arc 
as E V, is called the chord of the arc or 
segment E T V. The line T D bisecting 
the chord, is the versed sine, the line 
T II touching the circumference, and per¬ 
pendicular to the radius is called the tan- 
<; ent. The space enclosed between an arc 
and the two radii is a sector, as T V O. 

A line drawn perpendicular from the 
center of the Chord of any arc will give 
the Diameter; and if we draw lines from 
the centers of any two Chords, the inter 
section of those lines will be the center of 
the circle. 

A secant is a line, O H, that cuts a cir¬ 
cle, lying partly inside and partly outside 
of the circle. 



12 


THE CARPENTERS* AND JOINER’S 


A Semi circle, A 0 B. is one half of the 
circumference. A Quadrant, B F, Is one 
quarter of the circumference. 

A Sphere is a solid with a continued 
curved surface, equally distant at every 
point from the center. 

A Spheroid is a solid like a sphere, but 
not exactly round, one diameter being 
longer than the other. 

note. —The terms diameter, radius, seg. 
ment. &c. apply to spheres and spheroids 
the same as they do to plane surfaces. 

An ellipse is the section of a cone or 
cylinder cut by a plane passing obliquely 
through both sides. 

A cone is a solid having a circle for a 
base, and tapering to a point or vertex. 

Conic sections are the figures made by a 
plane cutting a cone. 

A parabola is the section of a cone cut 
by a plane parallel to its side. 

A hyperbola is the section of a cone 
cut by a plane at a greater angle through 
the base than is made by the side of the 
cone. 

A Cylinder is a solid having equal and 
parallel circles for its ends, and would be 
generated by the revolution of a rectangle 
about one of its sides. 


POCKET COMPANION 


VS 


A conoid is ii solid generated by revolv. 
mg a parabola or hyperbola on its axis. 

An arc is any part of a circle, as B F. 

The bask of any figure is that side upon 
which it is supposed to stand, and the alti. 
tude or height is the perpendicular falling 
upon that base from the opposite angle. 

The perimeter of any figure is the sum 
of all its sides taken together. 

In the ellipse, the longest diameter is 
called the transverse axis. The shortest 
diameter is called the conjugate axis. 

An ordinate is a line drawn from any 
point in tlie circumference parallel with 
one of the diameters. 

To draw an ellipse with a string. 



Let A B, pig. 5, equal the transverse 
axis, and 0 D the conjugate. Set the PL 
viders from A to O, and with the same 




14 


THE CARPENTER’S AND JOINER’S 


extent, from D as a center, cross the long, 
est diameter near each end at 1 and 2. 

Stick a brad awl at 1, another at 2, and 
one at D ; tie a string tight around them 
and then pull out the awl at D; place a 
pencil inside the stringy and by working 
it around, keeping the string tight, a very 
good Ellipse can be drawn. 


- • •• . X 

To strike an arc or segment through any 
three given points. 



Let A II O fig. 6, be the three points, 
with any radius less than the least dis. 
tance between either of the points, and 
with each of the three points as centers, 
describe arcs crossing each other, and 
through the points of intersection draw 
lines intersecting at D, which will be the 
center to strike from, by taking the dis. 
tance to any of the points as ;a radius. 




POCKET COM PANION. 


15 


To inscribe a Circle in any Triangle. 



Bisect two angles, ( See fig. 2, page 10 ) 
as A and B, fig. 7 ; then the intersection 
of the lines at O will be the center of the 
circle, and D O the radius. 


To find the length of any part of a circle. 



Divide the chord A B, into four parts; 
with one part as a radius cut the arc at O 




10 


THE CARPENTER'S AND JOINER'S 


draw O I) which will be as near one half 
the length of the arc A O B as can be 
drawn by any other method. 

To get the length of a Quarter Circle. 

Add together the chord and versed 
sine. ( See page 11. ) 


To describe an Elliptic Arch. 



Divide the Chord A B, pin. 0. into four 
equal parts; with each point of division 
for a center, and each part as a radius, 
describe three circles cutting each other 
in centers. Through the intersection of 
the outside points draw lines crossing at 
D which will be the center for striking 
the flat part, or top of Arch. 

A full Ellipse can be drawn by drawing 
two more lines intersecting opposite I) 
for another center. 





POCKET COMPANION. 


17 


To draw Lines leading to—or from— 
the Circumference of a Circular Arc, and 
tending toward the Center, when the Cen¬ 
ter is inaccessible. 



Divide the Arc into the required number 
of equal parts. Then with any radius 
less than two of these parts, describe arcs 
cutting each other as follows:— 

From A and 2 as centers cross at 4; 
from 1 and 3, cross at 5; from 2 and 11 
cross at C. Then draw 1 4, 2 5, and 3 6. 
To draw the End Lines, as at A and E 
With the same radius, and from 1 and 3 
as centers draw Arcs at 7 and 8; then with 
the radius 1 4, from A and E as centers, 
cross at 7 and 8. Draw 7 A and 8 E. 


The Radius of a Circle will go just six 
times around it. 




18 THE CARPENTER’S ANI) .JOINER’S 


To 44 Saw .kerf ” Boards, Moldings, &c. 
to bend around a Circular opening. 



Let A E B. fig. 11, represent a Circular 
opening, and O B a board to be bent on 
the outside of this circle. 

Make one cut in the board ( or any piece 
of the same thickness) to the required 
depth, and with the same saw to be used 
in making all the cuts. Now place the 
side of the board even with the diameter 
of the circle, and the saw .kerf at the center 
at O. I rom B set up the perpendicular 
B 11; stick a brad awl or nail at each side 
of the piece, as at 1 and 3, so as to keep 
this end stationary; then spring the piece 
slowly until the cut at O is just closed, 
and make a mark on the line H, where the 
face of the board comes to. This will be 
the distance between the saw-kerfs. 

Care must be taken so that the piece be 



POCKET COMPANION. 


19 


kept perfectly straight at each side of the 
saw-kerf while springing, else it is evi¬ 
dent too great a space will be obtained. 

To bend a piece on the inside of a circle 
deduct twice the thickness of the material 
from the width of the opening, and pro. 
ceed as above. 

note. —It is not necessary to lay out the 
Circle or Segment in full; just draw a line 
on any board, setting up a Perpendicular 
near one end, and let the distance from 
the saw .kerf to this perpendicular line be 
equal to the Kadi us. 

To get the distances for bending stuff 
around an Elliptic form; first find the two 
nearest radii—getting the space from each 
radius separately. ( See fig. 6, page 14,for 
for finding the radius of an arc. ) 

When it is desired to have a smooth sur¬ 
face on the face of panels or other work to 
be bent, a better, but more expensive way 
is, to “ dado ” the stuff on the back to one. 
quarter of an inch, or less, of the face; the 
distance between the gains may be equal 
to the size of the dado. The board is then 
fastened around a “ form ” of the desired 
shape, and keys fitted and glued into the 
gains until the piece fits the form true. 

In making circular Soffits, such as Heads 
for Window frames, Door frames &e. it 


20 


TIIE carpenter’s and joiner’s 


is cheaper to build up the circle out of one 
or more thickness of plank sawed to the 
proper shape. Sometimes these pieces are 
nailed together in the rough, and then 
veneered with some very thin boards. 


To gel the Stretchout of a Circle. 

Multiply the Diameter by the Decimal 
term 3.1416 ( or to be still more exact, by 
3.1415926) and the product will be the cir. 
cumference of the whole circle. 

The term, 3.1416 is the one generally 
used, being near enough to work to. 


To Describe a Circle that will 
be equal to two given Circles. 

Construct a Triangle, making the two 
Diameters of the given Circles two of the 
sides, forming a Right Angle. On the 
Hypothenuse of this Triangle for a Di. 
ameter, describe a Circle, the Area of 
which will be equal to the other two. 

A Square may be formed in the 

same manner equal to two other Squares, 
by making the Base and Perpendicular 




POCKET COMPANION. 


'21 


each equal to one of the sides of the two 
given Squares, and then constructing an. 
other Square on the Hypotiienuse, 


Angles, Polygons, &c. 

An Angle is the opening between two 
lines, meeting at a point, and is called a 
Right Angle, when the two lines are 
perpendicular to each other; an Acute 
Angle, when it is less or sharper than 
a right angle; and Obtuse, when it is 
greater than a right angle. 

A Triangle is a figure of three sides, 
and is called Equilateral, when all its 
sides are equal; Isosceles, when two of 
its sides are equal; and Scalene, when 
all the sides are unequal. 

Equilateral triangles are also equian¬ 
gular. Isosceles Triangles have two equal 
angles, as well as two equal sides; and sea. 
line triangles have the angles as well as 
the sides unequal. 

A Quadrilateral or Quadrangle is 
a figure of four sides, and when the four 
sides are all equal, and the angles are all 
Right Angles, it is called a Square; 
when the opposite sides are parallel and 
equal, it is called a Parallelogram. 



22 


THE CARPENTER’S AND JOINER’S 


When the sides are all equal, but the an. 
gles not right angles, it is termed aRhom. 
bus; when the opposite sides are equal, and 
the angles all right angles, it is termed a 
Rectangle; when the opposite sides are 
equal and parallel, but the angles are not 
right angles, it is called a Rhomboid; if 
all the sides are unequal, it is a Trapezi. 
dm ; and when only one pair of opposite 
sides are parallel, it is a Trapezoid. 

All plane figures having more than four 
sides are termed Polygons; and are called 
regular, when the sides and angles are all 
equal; and irregular, when they are not 
equal. They are named according to the 
number of their sides and angles, viz: 

A figure of five sides is called a pentagon. 


A 

do. 

“ six 44 

44 

a hexagon. 

A 

do. 

‘ 4 seven 44 

44 

a heptagon. 

A 

do. 

44 eight 44 

44 

an octagon. 

A 

do. 

44 nine 44 

44 

a nonagon. 

A 

do. 

44 ten 44 

44 

a decagon. 

A 

do. 

44 eleven 44 

44 

a undecagon. 

A 

do. 

44 twelve 44 

•4 

a dodecagon. 


note. — A triangle is sometimes called 
a Trigon, and a square a Tetragon. 

The Hypothenuse of a triangle is the 
side opposite the right angle. 


POCKET COMPANION. 


A Prism is a solid of three or more sides 
whose ends are equal, similar, and parallel, 
and whose sides are parallelograms. 

A Cube is a right prism having six equal 
square faces, all its angles right angles, and 
its opposite faces parallel. 

A Parallelopipedon is a prism hav¬ 
ing six quadrilateral faces or planes, the 
opposite pairs being equal and parallel. 

A Pyramid is a solid whose sides are all 
triangles, meeting at a common point 
called the vertex or summit , and whose base 
is a rectilinear figure, as a Triangle, 
Square, or Polygon, and from the shape 
of which the pyramid is named as Penta¬ 
gonal, Octagonal, &c. 

A Segment of a pyramid is a part cut off 
by a plane parallel to the base. A frustum 
or trunk of a pyramid is the portion re¬ 
maining after a part has been cut off par¬ 
allel to the base. 

A Prismoid is a solid having parallel 
ends or bases, dissimilar in shape, with 
quadrilateral sides. 

A Wedge is a solid of five sides, of which 
two are rhomhoidal , meeting at a common 
line or edge with a rectangular base, and 
triangular ends. 


24 THE CARPENTER’S AND JOINER’S 


To lay out an Octagon with 
A Framing Square. 

Suppose you have a stick of timber any 
size — from 1 inch to 24 inches square; lay 
the blade of the Square across one of the 
sides diagonally, so that the whole length 
of the blade will be on the stick, and the 
outside corners even with the opposite 
sides; make a mark at 7, and another at 17; 
set a Gauge to either of these points for 
taking oft* the corners. 


To describe an Octagon with the 
Compasses. 

First form a Square of the required di¬ 
mensions; draw diagonals from each cor¬ 
ner crossing each other at the center; set 
one leg of the Compasses at one corner of 
the square, and extend the other leg to the 
intersection of the diagonals at the center. 

With this radius , and from each corner 
as a center, describe arcs cutting the sides 
of the square; draw lines across the cor¬ 
ners from where the arcs cuts the sides. 

In getting out large heads or half-rounds 
‘by hand’, it is better to form them into 
octagons first. 



POCKET COMPANION. 


25 


Mensuration of Surfaces. 

1. To find the area of a square , rectangle , 
rhombus or rhomboid. 

Buie .—Multiply the length by the per¬ 
pendicular height or breadth, and the pro¬ 
duct will be the area. 

2. To find the area of a triangle. 

Buie .—Multiply the base by half the 

perpendicular height, and the product 
will be the area. 

Or, from half the sum of the three sides 
subtract each side separately; then multi¬ 
ply the half sum and the three remainders 
continually together, and the square root 
of the product will be the required area. 

3. To find the hypothenuse of a right- 
angled triangle. 

Buie .—Add together the squares of the 
two sides , and the square root of the sum 
will be the length of the hypothenuse. 

note. —By this rule the length of Rafters 
can be obtained, calling the Rafter the hy¬ 
pothenuse of the triangle; the rise, and 
half-width of Building, the two sides. 


26 THE CARPENTER’S AND JOINER’S 


4. To find the length of one of the legs when 
the hypothenuse and other leg is known. 

Buie .—Subtract the square of the known 
leg from the square of the hypothenuse, 
and the square root of the remainder will 
be the length of the other leg. 

5. The area and base of any triangle being 
known , to find the perpendicular height. 

Buie .—Divide double the area by the 
base, and the quotient will be the perpen¬ 
dicular height. 

Or, divide double the area by the height 
and the quotient will be the base. 

6. The base and perpendicular of any plane 
triangle being known, to find the side of its in¬ 
scribed square. 

Buie .—Divide the product of the base 
and perpendicular by their sum, and the 
quotient will be the side of its inscribed 
square. 

7. To find the area oj a trapezium. 

Buie .—Multiply either diagonal by half 
the sum of the two perpendiculars falling 
upon it, and the product will be the area. 

8. To find the area of any regular polygon. 

Buie .—Multiply half the perimeter by 

the perpendicular falling from its center 
upon either one of its sides, and the pro¬ 
duct will be the area. 


POCKET COMPANION. 


27 


9. To find the area of a trapezoid. 

Buie .—Multiply half the sum of the par¬ 
allel sides by the perpendicular height be¬ 
tween them; the product will be the area. 

10. To find the areas of irregular polygons. 

Buie. —Draw diagonals to divide the 

figures into trapeziums and triangles; find 
the area of each separately, and the sum 
of the whole will be the required area. 

11. To find the area of a ring: that is, the 
space enclosed between the circumferences 
of two circles, having a common center. 

Buie .—Square the diameter of each ring, 
and subtract the square of the less from 
the square of the greater. Multiply the 
difference of the squares by the decimal 
term .7854; the product will be the area. 

12 To find the area of any regular polygon , 
when the side only is known. 

Buie .—Multiply the square of the side 
by the multiplier opposite to the name of 
the polygon in the following table, and 
the product will be the area. 


28 THE CARPENTER’S AND JOINER'S 


Area , or 
Multipliers. 

0.433012 

1. 

1.720477 

2.598076 

3.633912 

4.828427 

6.181824 

7.694208 

9.365640 

11.196152 

Angle of 
Polygon. 

° « ^ 
ooocoooioo^b.0 

nnrinrinnri 

Angle. 

O ^ 

OONOrUOOOMO 

Name of 
Polygon 

Triangle. 

Square. 

Pentagon. 

Hexagon. 

Heptagon. 

Octagon. 

Nonagon. 

Decagon. 

Un decagon. 
Dodecagon. 

No. of 

sides. 



Additional use or this Table. 


The third and fourth columns of this 
table will be found useful in the construc¬ 
tion of those figures with the aid of the 
Sector or Plane Scale. Thus, if it is de¬ 
sired to describe an Octagon, opposite to 











POCKET COMPANION. 


29 


it, in the third column, is 45: that is, the 
side of an octagon forms an angle of 45 de¬ 
grees with a line extended outward from 
either of the other sides. 

On both the Sector, and Plain Scale, 
there is laid down a scale of Chords num¬ 
bered from 1 to 90,marked—c ho; to lay 
out an octagon with this Scale, first draw 
a base line to start from; set the compass¬ 
es from 1 to 60 and describe a part of a 
circle, using any point on this line for a 
center, one end of the arc also being on 
the line; then take the distance from 1 to 
45, place one foot of the compasses at the 
end of the arc resting on the base line, and 
describe another arc crossing the first one ; 
draw a line from this point to the first 
center used, and you will have two sides of 
the octagon. 

The length of those sides can then be 
laid out, and, having the angle, the other 
sides can be readily drawn. 

The fourth column gives the angle that 
any two adjoining sides of the different 
figures make with each other. 



30 


THE CARPENTER’S AND JOINER’S 


Arcs, Segments, Circles, Zones &c. 

1. The diameter of the circle, and versed 
sine of any arc being known, then, double 
the square root of the product of the differ¬ 
ence between the diameter and versed sine 
multiplied by the versed sine, will equal 
the chord. 

2. The square root of the sum of the 
squares of half the chord and of the versed 
sine equals the chord of half the arc. 

Or, the square root of the diameter mul¬ 
tiplied by the versed sine, equals the chord 
of half the arc. 

3. To find the versed sine . 

Buie .—Subtract the square of the chord 
from the square of the diameter, and ex¬ 
tract the square root of the remainder; 
deduct this root from the diameter, and 
one-half of the remainder will equal the 
versed sine. 

Or, from the square of the chord of half 
the arc, deduct the square oflialf the chord 
of the arc, and the square root of the re¬ 
mainder will equal the versed sine. 

Or, divide the square of the chord of 
half the arc by the diameter, and the quo¬ 
tient will be the versed sine. 


POCKET COMPANION. 




4. To find the diameter. 

Buie .—Add together the square of the 
versed sine and the square of half the chord 
of the arc, and then divide the sum by the 
versed sine. 

Or, divide the square of the chord of 
half the arc by the versed sine. 

note. —By this rule we can readily de¬ 
termine the* length of rod, or distance be¬ 
tween tram-points for describing circular 
segments. The rule gives the diameter of 
the circle, of which the arc is a portion. 

The difference between half the chord of 
the arc, and, the chord of half the arc is this: 
the first is one-half the width of the open¬ 
ing n£ the spring of the arch ; while the 
second is a line drawn from the highest 
point of the arc at the center, to either of 
its ends at the springing point. 

5. To find the length of an arc. 

When the number of degrees it contains, 
or the radius, chord and versed sine are 
known. 

Buie .—Multiply the number of degrees 
in the arc by the decimal term .0174533, 
and that product by the radius. 

Or, as the number of degrees in the cir¬ 
cle (360) is to the number of degrees in the 
arc, so is the circumference of the circle to 
the length of the arc. 


32 THE CARPENTER’S AND JOINER’S 


Or, from 8 times the chord of half the 
arc, subtract the chord of the whole arc, 
and divide the remainder by 3; this will 
be the length—nearly. 

Or, from the square of the chord of half 
the arc, subtract the square of half the 
chord of the arc, the remainder will be the 
versed sine. Then proceed as above. 

6. To find the area of a sector of a circle. 

Buie .—Multiply the length of the arc by 

half the radius. 

Or, as 360 is to the number of degrees in 
the arc, so is the area of the whole circle to 
the area of the sector. 

7. To find the area of a segment of a circle. 

Buie .—If less than a semi-circle, And the 

area of a sector having the same arc as the 
segment, and deduct therefrom the area of 
the triangle formed by the radii of the sec¬ 
tor and the chord of the arc. 

If greater than a semi-circle, find the 
area of the lesser portion of the circle by 
the above rule, and deduct it from the area 
of the whole circle. 

8. To find the area of a zone. 

Buie .—Deduct the sum of the areas of 
segments formed on the parallel sides of 
the zone from the area of the whole circle. 


l’OOK l'/f COMPANK )\. 


33 


Or, to the area of the largest trapezium 
or parallelogram formed inside the zone, 
add the areas of the segment at each end 
of the parallelogram or trapezium. 

9. To find the area of a lune or crescent. 

Rule. —Find the areas of the two seg¬ 
ments forming the lune, the difference will 
he the area required. 

10. To find the area of an ellipse. 

Rule. —Multiply its longest axis by its 
shortest, and its product by .7854. 

11. To find the circumference of an ellipse. 

Rule. —Square its two axes, and multiply 

the square root of half its sum by 3.1416. 

12. To find the area of an elliptic segment , 
cut off by a line perpendicular to either axis. 

Rule. —Find the area of a corresponding 
circular segment of equal altitude, and of 
same vertical axis or diameter. Then, as 
the vertical axis is to the one parallel 
with the base, so is the area of the circular 
segment to that of the elliptic segment. 



34 


THE CARPENTER’S AND JOINER'S 


Mensuration of Solids. 

1. To find the superficies or outside surface 
of a cylinder or prism. 

Buie .—Multiply the perimeter (distance 
around) of the base by the altitude, the 
product will be the lateral surface; then 
add to this the area of the bases or ends 
for the whole surface. 

2. To find the solid content of a cylinder or 
prism. 

Buie .—Multiply the area of the base by 
the height. 

3. To find the outside surface of a cube. 

Buie .—Add together the areas of the sev¬ 
eral sides. Or, multiply the girt, or peri¬ 
meter of the figure by the length of the 
sides, and to the product add the area of 
the ends or bases. 

4. To find the solid content of a cube. 

Buie .—Multiply the area of the base by 

the height. 

5. To find the outside surface of a regular 
pyramid or of a cone. 

Buie .—Multiply the perimeter of the 
base by half the height of one side, and add 
the area of the base when required. 


POCKET COMPANION. 


35 


For a frustrum of a pyramid or cone, 
t ake half the sum of the perimeters of the 
the two ends by half the slant height. 

6. To find the solidity of a pyramid or cone. 

Rule .—Multiply the area of the base by 

one-third the perpendicular height. 

7. To find the solidity of a frustrum of a 
pyramid or cone. 

Rule .—To the area of the two ends add 
the square root of their product, and this 
multiplied by one-third the perpendicular 
height will give the solidity. 

8. To find the solidity of a wedge. 

Rule .—To the length of the edge add 
twice the length of the base; then multi¬ 
ply the sum by the height of the wedge, 
and the breadth of the base, and one-sixth 
of the product will be the solid contents. 

9. To find the so lidity of a rectangular pris- 
rnoid. 

Rule .—To the areas of the two ends add 
four times the area of the middle section, 
then multiply the sum by one-sixth of the 
height; the product will be the solidity. 

The surface of a sphere is to the whole 
surface of a circumscribed cylinder, its 
bases included, as 2 is to 3, and the solidi¬ 
ties of these bodies are in the same propor¬ 
tion. 


36 


THE CARPENTER'S AND JOINER’S 


10. To find the surface of a sphere. 

Buie. —Multiply the circumference by the 
•diameter, or multiply the square of the di¬ 
ameter by the decimal term 3*1416. 

11. To find the solidity of a sphere. 

Buie. —Multiply the cube of the diameter 
by the decimal *5236. 

12. To find the surface of a spher ical seg¬ 
ment. 

Buie. —Multiply the whole circumference 
of the sphere, of which the segment is a 
portion, by the height of the segment. 

13. To find the solidity of a segment. 

Buie. —To three times the square of the ra¬ 
dius of its base, add the square of its height, 
then multiply this sum by the height, and 
that product by the decimal *5236. 

Or, from three times the diameter of the 
whole sphere deduct twice the height of the 
segment, the remainder multiplied by the 
square of the height and the product by the 
decimal *5236 will give the solidity. 

14. To find the surface of a ring. 

Bale .—Multiply half the sum of the outer 
and inner circumference by the girt. 

15. To find the solidity of a ring. 

Buie. —Multiply half the sum of its outer 
and inner circumference by the area of its 
cross section. 



POCKET COMPANION. 


o 7 


-MEMORANDA- 

Concerning Various Materials. 

Air. Specific gravity, 00012; weight of a 
cubic foot, 527 grains; 13*3 cubic feet of 
air will weigh 1 lb. 

• Ash. Specific gravity 0*70; weight of a 
cubic foot, 47*5 It'S; weight of a bar one foot 
long,'and one inch square, 0*33 Its. ; will 
bear without permanent alteration a strain 
of 3,540 It s. upon a square inch. 

Atmosphere. The pressure of the atmos¬ 
phere is nearly 15 lbs. upon a square inch, 
and is eqal to a column of water 34 ft. high. 

Beech. Specific gravity, 0-096; weight of 
a cubic foot, 45*3 lbs. ; will bear without 
permanent alteration on a square-inch, a 
pressure of 2,360 lbs. 

Brass , cast. Specific gravity, 8*37 ; a 
cubic foot weighs 523 lbs. ; weight of a bar 
one foot long and one inch square, 3-63 lbs. 
melts at 1,8090 ; will bear on a square 
inch without permanent alteration, a press¬ 
ure of 6,700 lbs. 

Brick. Specific gravity, 1*841 ; weight 
of a cubic foot, 115 lbs. , and will absorb 
about % of a gallon of water; is crushed 
by a force of 562 lbs. upon a square inch. 




38 THE CARPENTER’S AND JOINER’S 


Bridges. When an ordinary Road bridge 
is covered with people, it is about equiva¬ 
lent to a load of 120 lbs. to a square foot. 

This may be safely calculated on as being 
the greatest load with which the bridge is 
likely to be covered , and one incapable of 
supporting such a load cannot be deemed 
safe. 

Cast Iron. Specific gravity, 7*207; a cu¬ 
bic foot weighs 450 lbs. ; a bar one inch 
square and one foot long, w*eighs nearly 
3*2 lbs. ; melts at 3,479o ; and shrinks in 
cooling about of an inch for each foot in 
length; is crushed by a force of 93,000 lbs. 
upon a square inch, and will bear without 
permanent alteration 15,300 lbs. upon a 
square inch. 

Coal , Newcastle. Specific gravity,, 1*269; 
weight of a cubic foot, 79*31 lbs. ; a bushel 
is generally rated at 84 lbs. 

Copper. Specific gravity, 8*75; weight of 
a cubic foot, 549 lbs. ; weight of a bar one 
inch square and one foot long, 3*81 lbs. ; 
and melts at 2,548 O . 

Earth , common. Specific gravity, 1*52 to 
2 00; weight of a cubic foot, from 95 to 125 
lbs. ; a cubic yard of common soil weighs 
about 3,429 lbs. 


POCKET COMPANION. 


39 


Granite. Specific gravity, 2*625; weight 
of a cubic foot, 164 lbs. ; is crushed by a 
force of 10,900 lbs. upon a square inch. 

Gravel. A cubic foot weighs about 120 lbs. 

Gun Metal. ( Copper 8 parts, tin 1 part ) 
Specific gravity, 8*153; a cubic foot weighs 
50934 lbs. ; weight of a bar one foot long and 
one inch square, 3*54 lbs. ; melts at 2,500o ; 
and shrinks in cooling about one-sixth of an 
inch for each foot in length. 

Iforse-Poicer. The unit of nominal power 
for steam-engines, or the usual estimate of 
dynamical effect per minute of a horse, call¬ 
ed, by engineers, a horse-power , is 33,000 lbs 
at a velocity of 1 foot per minute; or, the 
effect of a load of 200 lbs., raised by a horse 
for 8 hours a day, at the rate of 234 miles 
per hour, or 150 lbs, at the rate of 220 feet 
per minute. 

Lead , cast. Specific gravity, 11*353; a cu¬ 
bic foot weighs 70934 lbs. ; weight of a bar 
one foot long and one inch square, 4*94 lbs. 
and melts at 6120. 

Slate. Specific gravity, 2*752; weight of 
a cubic foot, 172 lbs. 

Steel. Specific gravity, 7*84; weight of a 
cubic foot, 490 lbs. ; weight of a bar one 
inch square and one foot long, 3*4 lbs. 

Water, fresh. Specific gravity, 1* ; acu- 


40 THE CARPENTER’S AND JOINER’S 


bic foot weighs 6234 lbs. ; weight of a gall¬ 
on, 10 lbs. ; expands in freezing, about one- 
seventeenth of its bulk; and the expanding 
force of freezing water is about 35,000 lbs. 
upon a square inch. 

Water, sea. Specific gravity, 1*0271; a 
cubic foot weighs 64*2 lbs. 

Wind. Greatest observed velocity, 159 ft. 
per second; force of wind with that veloci¬ 
ty, about 57% lbs. upon a square foot. 


-NAILS_ 

Number per pound of different 
Kinds and sizes. 

3 penny, fine, 1 y z long, 720 in a pound. 


3 

44 

common 

, 13 £ 4 * 

400 

LL 

4 

a 

<( 

1% 44 

300 

LL 

5 

ll 

a 

m “ 

200 

LL 

6 

ll 

44 

2 in. 44 

150 

LL 

7 

ll 

44 

2 % “ 

110 

LL 

8 

44 

LL 

2% 44 

85 

44 

9 

44 

44 

2% 44 

75 

44 

10 

44 

LL 

3 in. 44 

60 

LL 

12 

44 

lL 

3% “ 

50 

LL 

16 

ll 

LL 

3% 44 

40 

LL 

20 

ll 

LL 

4 in. “ 

22 

LL 

30 

44 

LL 

4% “ 

16 

44 

40 

ll 

LL 

5 in. “ 

12 

44 





POCKET COMPANION. 


41 


Table of Nails, continued. 

50 penny, com. 534 in. long, 10 in a 11). 

60 “ “ 6 “ 8 1 

Casing Nails. 

6 penny,_2 in. long. 220 in a pound. 

8 “ _2^ “ 125 

10 “ _3 in. “ 85 “ 

12 “ ....3 “ 63 “ 

Fence Nails. 

6 penny, .... 2 in. long, 80 in a pound. 

8 “ _ 234 “ -">0 

10 “ _3 in. “ 30 “ 


From 434 to 5 lbs. of 4 penny nails, or 3 
to 334 lbs. of 3 penny nails will lay 1000 
shingles. 5% lbs. of 3 penny tine lath nails 
will put on 1000 laths, joists being 16 inches 
to centers. 234 lbs. of 8 penny common 
nails will lay a square of flooring of 534 in. 
matched boards, joists being 16 inches to 
centers. 234 lbs. of 7 penny nails will put 
on a square of lap siding, showing 4 inches 
to the weather. 




42 


THE CARPENTER'S AND JOINER'S 


TIIE FRAMING SQUARE. 

As a general rule, Carpenters are not 
thoroughly familiar with the marks on the 
Framing Square. To enumerate in detail, 
what might be accomplished with it, would 
require a good sized volume devoted ex¬ 
clusively to the subject. 

Some of the following explanations may 
appear tedious to the advanced workman, 
but it is for those who need information 
that it is intended. 

Let us now take up the “ Eagle Square, 
No. 1 ” and see what we can find on it. 

The Board Rule. 

Across one side of the Blade at 12 inches, 
is laid down the following figures, commen¬ 
cing at the bottom : 8,9, 10, 11, 12, 13, 
14 and 15; these numbers represents the 
length in feet of the board to be measured, 
and also the number of feet in a board 12 
inches wide. For any other width, as 8 in¬ 
ches, and the length 15 feet, opposite 8 in 
the space marked 15 is 10, the number of 
feet in the board. If the board is shorter, 
or longer than is given on the Square, take 
one-half or double the numbers. 


POCKET COMPANION. 


43 


The Brace rule. 

On the Tongue of the Square, and on the 
same side that contains the board rule, is 
the Brace rule . The figures are laid down 
as follows: 24 ^ 27 38 . 19 &c . 

The double numbers represent the two 
sides of a square, and the decimal numbers 
are lengths of diagonals. Thus, if the two 
sides of a square are 24 each, the length for 
a brace will be 33 and 9 tenths; if the sides 
are 27, the brace will be 38 and 19 hundred¬ 
ths, and so on. 

Tiie Scale of Diagonals. 

The Diagonal Scale is laid down at the 
end of the ‘brace rule’ near the cor¬ 
ner of the Square, which consists of eleven 
equi-distant parallel lines, crossed by ver¬ 
tical ones, drawn obliquely, as follows:— 

One inch is divided into ten equal parts 
above and below, and a line drawn from the 
top of the first perpendicular to the first di¬ 
vision below, and continued parallel, by 
which method the first division on the sec¬ 
ond line from the top becoms 100th of an 
inch, and each of the others on the parallel 
lines one tenth. 


44 


THE CARPENTER’S AND JOINER’S 


The scale of hundredths reads downwards 
from the second parallel line where it inter¬ 
sects the first diagonal at the right-hand 
corner, this being 100th of an inch. The 
next space below is 200ths,the next 300ths, 
and so on to the last, which is 10 hundred¬ 
ths, or one-tenth of an inch. 

The Octagon Rule. 

On the other side of the Square along the 
Tongue , are a series of numbers : 10, 20. 30. 
40,50, and 60; which are five-twelvths of 
one-half as many inches, being the tangent 
of 22340 for circles whose diameters are 10 
20,30 &c. inches. These numbers are used 
as follows: We have a stick of timber 10 
inches square, and wish to make it eight 
square; draw a center line on each side of 
the stick; set the dividers from the begin- 
ing of the Scale to 10. and set off this dis¬ 
tance on each side of the line; this shows 
how much of the corner to take off. 

The Octagon Scale is also on most of the 
Pocket Rules, given on two lines, marked 
M and E. The divisions on the line, M 
will be found to correspond with those on 
the Square, and are used in the same way. 

The line, E, gives the distance from the 
corner of the stick to the angle of the octa- 


POCKET COMPANION. 


45 


gon, and is very convenient. Suppose you 
have a stick 6 inches square; set a Guage 
to 6, on the line, E, and guage this quanti¬ 
ty each way from all the corners. 

The lines, M, and E, can also be used for 
'A Brace Buie; the line, E, being a table of 
of equal sides of a square, and M, a table of 
diagonals, or Braces. 


Method of applying the Square 
On different hinds of work. 

To lay out a Common Rafter. 

A Roof is said to be a “ quarter pitcli ” 
when the rise of the rafter is the width 
of the building; a half\ or square pitch, 
w hen the rise is^ the width, &c. 

The figures, 6 and 12, (or any other pro¬ 
portionate numbers) will give thfe cuts for 
a quarter pitch ; 8 and 12 for a third pitch ; 
12 and 12 for a half pitch; from which it 
will be observed that the rise per foot of 
the several pitches are, for a quarter pitch 
6 inches; a third pitch 8 inches; a half pitcli 
12 inches. Other pitches are often used, 
as 3, 5, or 7 inches rise per foot. To get the 
cuts for those, use 3 and 12, 5 and 12, 7 and 



46 


THE CARPENTER'S AND JOINER'S 


12, or whatever it may be. To “layout” a 
common rafter for, say a quarter pitch roof; 
suppose we begin at the foot, placing the 
figures, 6 and 12 on the outside edge of the 
Square even with the edge of the scantling 
and mark the long bevel from 12; this gives 
the “heel-’ cut which sets on the plate. To 
get the “plumb” cut for the outside edge of 
the plate, draw a line square from the heel, 
leaving at least one-half the width of the 
rafter solid over the plate. 

To get the Length. 

Extend the plumb cut to the edge of the 
scantling forming the back, or upper edge 
of the rafter. Set 12 against the end of this 
line, and 6 even with the back; then, with 
the Square in this position, space off as 
many lengths as there are feet in half-width 
of the Building, marking the top cut at the 
last space, using the bevel obtained at 6. 

Should a fraction of a foot occur in half 
the width of the Building, lay out the last 
space in full for even feet; then slide the 
Square back on the long bevel until the in¬ 
ches on the Square at this bevel shall corres¬ 
pond with those of the width. 

Some add 134 inches to each foot of half 
width for a quarter pitch, but this makes 


POCKET COMPANION. 


47 


the rafter a trifle long, making but little 
difference on a short rafter, being about % 
inch too long for each 8 feet in length; a 
little practice with the Framing Square 
will soon be found more practicable than 
any other method. 


HIP RAFTERS. 

To get the plumb and heel cuts for a Hip, 
use the same figures for the rise that you 
do for the Common rafters of like pitch, 
but instead of taking 12 inches for the run 
take 17 —providing it is a square run; this 
number being the diagonal of a square, the 
sides being 12. Therefore, for a quarter 
pitch roof, 6 and 17, will be the bevels for 
the heel and plumb cuts. For any other 
pitch, use whatever the rise may be for a 
common rafter, but always take 17 for the 
run. 

To get the Length of the Hip. 

With the Square at 6 and 17 (for % pitch) 
space oft* as many lengths as you would for 
common rafters on the same plan;thus, for 
a quarter-pitch Building 30 ft. square, the 
hips running to the center space off 15 times 
with the Square as above, viz : 0 and 17. 



48 THE CARPENTER'S AND JOINER'S 


When the Hip is Out of Square. 

Suppose you have a rectangular plan, say 
16 x 24 ft. and want one or more hips run 
to the center. Halve each of the adjoining 
sides, and we have 8 ft. and 12 ft. respect¬ 
ively for runs. Let the total rise of the 
rafter be, say 6 ft. ; then the pitch on the 
16 ft. side will be 6 ft. rise and 12 ft. run, 
and the 24 ft. side will be 6 ft. rise and 8 ft. 
run. Therefore, 6 and 12 will give the cuts 
for the common rafters on the sftortfside, and 
12 spaces with the square in this position 
will give the length; and 6 and 8,—or its 
equivalent 9 and 12—will give the cuts for 
the common rafter on the long side, and 8 
spaces with the Square at 9 and 12 will 
give the length. 

To get the Hips. 

Measure the distance from 8 to* 12 across 
the corner of the Square, and it gives 1434. 
Then, 6 and 1434 011 the Square will give the 
cuts, and 12 spaces with the Square in this 
position will give the length. 

It makes no difference with this method 
how much run there is either way; suppose 
you have 6 ft. run on one side, and 734 ft. 
on the other; lay down the Square to 6 and 


POCKET COMPANION. 


49 


7^, and mark the long bevel; slide the 
Square along this line to 12, and the other 
side will be at • measure across the cor¬ 
ner of the Square from 9% to 12, and it 
gives 15J4. Then, the rise per foot of the 
6 ft. run, with 15^ will give the cuts for the 
Hip, and 73^ spaces with the Square in this 
position will give the length. 

Valley Rafters. 

Valley rafters are obtained on the same 
principle as Hips of corresponding rise and 
run, the top being mitered from the center 
each way to fit the ridge or other rafters. 

Jack Rafters. 

To get the length of Jack rafters, take as 
many spaces with the Square as there are 
feet from the corner to where the jack sets, 
using the same figures that you do for com¬ 
mon rafters of the same pitch. 

To get the Side Bevel. 

Measure the diagonal of the rise of one 
foot run of the common rafter, measuring 
across the Square from 6 to 12 (for a quar¬ 
ter pitch,) which will be nearly 13*^ inches. 
Now lay down the Square to 12 and 1334> 


50 


THK CARPENTER'S AND JOINER'S 


the same as in laying out a miter, and use 
the long bevel. The down bevel will be the 
same as that of the common rafter. 

Jack rafters should be got out in pairs, 
sawed true to the bevels, and numbered; 
the pair next the corner being No. 1. 

If the preceding instructions are strictly 
adhered to, no difficulty need be anticipa¬ 
ted as to their not being right 


CRIPPLE STIJOS. 

Cripple studs vary in length in regular 
order for any pitch. If the roof is a quar¬ 
ter pitch, with studding 12 inches from 
centers, each stud would be 6 inches lon¬ 
ger than its next shortest one, being a rise 
of 6 inches per foot. Where they are 16 in¬ 
ches from centers, first lay down the Square 
to 6 and 12, marking the side to 12; slide 
the Square along this line to 16, and on the 
tongue will be found the rise—8 inches, the 
difference in length of the studs. 

To get the side bevel, use same figures as 
above, the lengths and bevels varying for 
the different pitches. 




POCKET COMPANION. 


51 


To get a Bevel for obtaining a 
Miter across tlie edge of'a 
Bevelled piece. 

Suppose you have a board % of an inch 
thick, and bevelled on % of an inch, and to 
a feather-edge at the back; the bevel for the 
miter will be obtained as follows : 

Lay down the Square ( the same as in 
laying out a rafter, ) with the figures, 3 in. 
and 7 in. even with the edge of a straight 
board, and draw a line from 7 to the cor¬ 
ner of the Square; slide the Square along 
this line to 12, and the other side will be 
at 5%; measure across the corner of the 
Square from 12 to 534> anf l it gives about 
13 inches; then, lay down the Square to 
12 and 13, and set a bevel to the short line; 
this gives the miter across the bevelled 
edge. 

For any other bevel than the one above 
given, proceed according to the following 
General Rule. 

Buie .—Take as many inches each way on 
the Square, as will correspond to the num¬ 
ber of 8ths, 12ths, or 16ths that the piece is 
bevelled each way; or one-half the number 
or double the number, so that the ratio 


52 


THE CARPENTER'S AMI) JOINER'S 


of the numbers will not be changed; draw 
a line along either the Tongue or Blade , 
and slide the Square along this line until 
either side reaches 12 inches first; measure 
the diagonal from 12 to where the other 
side will be, and use this diagonal distance 
with 12, to get the miter,—using the short 
bevel. 


To miter two Pieces together 
when both stand out of plumb, 
such as Splayed or Flaring 
Boxes, Hoppers &c. 

First find the pitch that the sides and 
ends will stand on, viz : 3 inches per foot, 
6 inches per foot, or whatever it may be. 

Suppose the sides and ends of a box stand 
six inches out of plumb for each foot of per¬ 
pendicular height; then, the diagonal of 6 
and 12 (13^) will be the figures to use 
with 12 to get the miter, laying the Square 
across the square edge of the pieces, to 12 
and 13using the short bevel. 

Where the edges of the pieces are to be 
bevelled, so as to be parallel with the base, 
a square miter will do; and also, by apply¬ 
ing a Try-Square across this bevelled edge, 



1‘OCKKT COM TAN ION. 


and then marking the pitch on both sides 
of the piece, a butt-joint will be obtained. 

Where the sides form a different angle 
from the ends:— To get the miter . 

First, get the miters for each pitch separ* 
ately by the preceding methods, and draw 1 
them both across the edge of one of the 
pieces, the point of intersection forming an 
acute angle; bisect the angle thus formed, 
which will be the required miter. 


laiftTS OI S€ROLL-»RAWIAG. 



The art of Scroll Drawing when done 
with an ordinary pair of dividers, or com¬ 
passes, consists merely in finding the right 
points to strike from, so that the change in 
the curvature of the arc will not be abrupt. 

That a true taper-scroll cannot be drawn 
with the Compass is evident, as, in that 
case, the radius should be constantly chang¬ 
ing ; while, for ordinary purposes, the 




the carpenter’s and joiner’s? 


54 


Compass will be found far more convenient 
than anything else. 

-In the figure on the preceding page, 
an elaborate design is dispensed with, as 
it would require such a great number of 
lines to show the several radii, that it 
would be more incomprehensible than a 
simple illustration, showing at a glance, 
the manner of obtaining a radius for any 
part of a circle. 

Draw a line, A B, and from any point 
on this line, with any radius, describe a 
half-circle. If you wish to continue the 
curve with a less radius, take any other 
point on this line that will be nearer to the 
point you start from again. To turn the 
curve in an opposite direction, take any 
point on the line, A B, outside of the arc 
for a radius. A very good Scroll can be 
drawn by w r hat is termed, “ half circles 
drawn to centers ” as follows: 

First draw 7 a straight line to work to, and 
on this line describe a half-circle. With 
a new radius on this line equal to one-half 
the former radius, describe another half 
circle opposite the first one; take a new r 
radius again on this line equal to half the 
last radius; and so on, as far as you wish. 


POCKET COMPANION. 


If it is required to draw another Scroll 
outside this one. so as to form a parallel 
distance between the two, use the same 
centers to strike from as at first, increasing 
the first radius to suit the required width. 

.With a little practice, a great variety of 
Scroll-work can be drawn with the Com¬ 
passes. If a Scroll is commenced with a 
half-circle, it should be continued with 
half-circles; if begun with a quarter-circle, 
let the whole be composed of quarter-cir¬ 
cles. In all attempts at Scroll-drawing 
with the Compasses, always bear this in 
mind:— Draw a line from where you wish to 
end any part of an Arc , to the last center used, 
and take a new radius anywhere on that line. 

If you wish to change the direction out¬ 
ward, or opposite, extend the line through 
and beyond the are, and take.a new radius 
at any point on that line. 

A continued Parallel Scroll consist¬ 
ing of half-circles, can be drawn from two 
centers, on a given line, the distance be¬ 
tween the centers being equal to the space 
between the lines of the Scroll; drawing 
each part only as far as the base line. 

A very good Taper Scroll can be 
drawn as follows:— Describe a small 


5(j 


THE CARPENTER'S AND JOINER'S 


quarter-circle, and draw lines from each 
end of the arc through, and beyond the * 
center described from. On either of those 
lines set back any distance, ( the greater 
the distance between the centers, the more 
spread there will be to the Scroll, ) and 
describe another quarter, drawing another 
line again from the last extremity of the 
arc through this last center. Set back on 
this line again, as far as on the previous 
one, and continue, by quarters, as far as 
desired. 

REMEMBER, that these three points 
must always be in a direct line, viz : The 
extreme end of the last Quarter Circle ,— The 
last Center used ,— And , the next Center to he 
used for a new radius. 

This is the whole secret of connecting 
Arcs of different radii, with the Compasses, 
and is something that every Mechanic can, 
and should be thoroughly familliar with. 

On the following page, is given a method 
of tracing a Scroll with a string, and can 
be relied upon as being acurate. This is 
nothing new, but nevertheless, is of great 
importance where a true Scroll is required 
to be drawn. 


POCKET COMPANION. 


57 


A TRI E TAPER-SCROLL 

Can be drawn by the following method: 
Make a Stock, or Center, in the form of an 
inverted cone, ( A Boy’s Spinning Top, for 
example, ) and fasten to a board, or pat¬ 
tern by means of a long spur placed in the 
small end. Wind a string around the stock 
commencing at the large end. The end of 
the string should have a loop, in which to 
place a pencil; cut a groove around the 
point of the pencil, to allow the string to 
turn freely without slipping off; place the 
pencil in the loop, and commence to un¬ 
wind the string from the Stock, until a 
Scroll of sufficient dimension is obtained. 

Any form of Scroll may be produced, by 
varying the size, and form of the Stock 
around which the string is wound. 

To trace a double line, by which a slot 
may be cut, it is only necessary, after des¬ 
cribing the first curve, to turn the Stock a 
little, and repeat the operation, thus des¬ 
cribing a second line parallel with the first 
one, and close to it. 


THE CARPENTER'S AND JOINER’g 


A PARALLEL SCROLL 

Can be drawn by using a Stock in the 
shape of a Cylinder. ( a round stick, ) for 
the string to be unwound from; even a 
square Stock, will give an approximate 
Scroll. A Scroll may be changed consider¬ 
able, by the way in which the string is 
wound around the Stock,—either close, or 
extended. A little practice at such work 
occasionally during leasure hours, is abso¬ 
lutely necessary, so that the principles of 
operation, and the several modes of applv- 
ing them, may be permanently fixed in the 
mind. 


PROPORTIONS of MOULDINGS, 
PROJECTIONS, &c. 

THE SCOTIA, OR COVE. 

Divide the width and thickness into four 
equal parts each, and take three parts for a 
radius, to describe the quarter-circle that 
forms the cove. 

THE HALF-ROUND. 

When used in connection with the Scotia 
as cap, or a solid moulding, its width should 
be the same as that of the Scotia. 



POCKET COMPANION. 


50 


BEVEL FOR CASINGS. 

The usual proportion for nearly all bev¬ 
elled work, such as Door Casings, Base 
for Columns, Piazza skirting &c., is, to 
Guage on one-half the thickness of the ma¬ 
terial, and work to a feather-edge at the 
back. 


PROJECTION OF CAPS. 

The greatest projection beyond the Bed 
Moulding for Caps to Columns, Wainscot¬ 
ing, outside Door and Window Frames &c., 
ought not be less than the thickness of the 
Cap at its projecting edge. 

Treads for Steps and Stairs, should only 
project one-half their thickness beyond the 
greatest projecting members under them. 

PROJECTION OF CORNICES. 

It is next to an impossibility to give 
any general rule to apply to the projections 
for Cornices, there being nearly as many 
different ideas in regard to it. as there are 
Builders. 

For Dwelling Houses, a projection of 
134 inches per foot of height, from Sill to 
Plate, is the one now most in use. 

The projection should be measured on a 


THE CARPENTER’S AND JOINER'S 


60 


line at right-angles to, or square from the 
Frieze to the outer edge of the Crown 
Moulding, both for the Eave and Rake. 

This rule applies chiefly to two-story 
houses. Where the Building is only one 
story in height, the projection ought not 
be less than 2 inches per foot; except such 
minor appurtenances as, Piazzas, Bay 
Windows &e., which can be 1^ inches per 
foot. 

WIDTH OF FRIEZE. 

For an ordinary Frame dwelling, the 
Frieze may be from 12 to 15 inches wide 
for the Eave Cornice, the Rake Frieze con¬ 
forming itself to that of the Eave, and 
varying in width according to the pitch of 
the roof. 

The greatest width of plain Frieze for a 
Piazza may be from 8 to 10 inches, except 
when It is to be arched, in which case, it 
can be from 12 to 18 inches from the foot 
of the arch to the Planceer. Those are 
generally made of two members, called, 
the inside, and outside Frieze, (except 
when the Columns are made of one thick¬ 
ness of, plank,) packed out with furring 
between them, and cased on the under side. 
This Casing, or under side of the Frieze is 


POCKET COMPANION. 


<;i 


termed the Soffit ” [ pronounced, sot-feet. 

The Soffit,—which includes the thickness 
of the inside and outside Frieze,— should 
be the same width as the Shaft, or body of 
the Columns. 

The Frieze for a Bay Window may be 
16 inches wide, having a Quarter-Round 
or other Moulding stuck on the lower edge 
where it laps on the Window Frame. 

Wide, plain Friezes on any Cornice, 
should be suitably ornamented with pan¬ 
els or scroll-work of tasty design. ’‘Mock 
Panels” are often made, by planting on 
some Half-Round Moulding, the ends of 
the panels being either square, octagonal, 
or pointed. The latter style is chiefly 
used on Gothic Buildings. 

When Bracketts are to be used, instead 
of having the Frieze wide enough to re¬ 
ceive them, it is often widened only where 
they are to be placed, showing a margin of 
five or six inches on each side of the Brack¬ 
ett. In this case, it looks well to run an 
Architrave Moulding along the lower edge 
of the Frieze, breaking down, and around 
the feet of the Bracketts. 


THE CARPENTER’S AND JOINER'S 


62 


FRAMING. 

The following Rules and explanations 
have been slightly abridged from “ Hat¬ 
field’s Carpenter’s Assistant” 

“Before proceeding to consider the 
principles upon which a system of fram¬ 
ing should be constructed, let us attend to 
a few of the Elementary laws in ‘Mechan¬ 
ics’, which will be found to be of great 
value in determining those principles. 

Laws of Pressure. 

“ A heavy body always exerts a pressure 
equal to its own weight in a vertical di¬ 
rection. Thus,—suppose a weight, W, in 
Flo. 13, weighing 100 lbs., be suspended 


n 



V 




POCKET COMPANION 


03 


from an iron rod, D B, then, of course, the 
pressure exerted upon the rod, will be 
equal to that of the weight viz : 100 Its. But 
if two inclined posts, A B, and B C, re¬ 
ceives the strain, then, the united pressure 
upon these posts will be more than equal 
to the weight — being in proportion to 
their position; the farther apart their 
feet are placed, the greater will be the 
pressure, and vice versa. Hence, tremend¬ 
ous strains may be exerted by a compara¬ 
tively small weight. 

And it follows, therefore, that a piece of 
timber intended for a strut or post, should 
be so placed that its axis may coincide, as 
near as possible, with the direction of the 
pressure. 

The direction of the pressure of the 
weight, W, is in the vertical line, D B, 
but, it exerts a proportionally greater 
pressure upon the oblique supports, A and 
C. It becomes important, therefore, to 
know the exact amount of pressure any 
certain weight is capable of exerting upon 
oblique supports. This can be ascertained 
by the following process: 

“Let A B, and B C, fig. 13, represent 
the axis of two sticks of timber supporting 
the weight, W; and let the weight, W, be 


04 


THE CA li PENT Eli’S AN1> JOlKKIi'S 


equal to 0 tons. Make the vertical liiie, 
B D, equal to 6 inches; from I), draw D F 
parallel to A B. and draw D E, parallel to 
B C; then, the line, B E, will be found to 
be, say, 4 inches long, which is equal to 
the number of tons that the weight exerts 
upon the post, A B. The pressure upon 
the other post is represented by B F, 
which in this case, is supposed to measure 
3 inches, representing 3 tons. 

Thus it will be found that the weight, 
which weighs only G tons, exerts a press¬ 
ure of 7 tons; the amount being increased 
because of the oblique position of the posts. 

The lines, E B, B F, F D, and D E, 
comprise what is called the parallelogram 
of forces. The oblique strains exerted by 
any one force, therefore, may always be 
ascertained by making B I), equal (upon 
any scale of equal parts ) to the number of 
lbs., cwts., or tons contained in the weight, 
W, and B E will then represent the num¬ 
ber of lbs., cwts., or tons with which the 
timber, A B, is pressed, and B F, that ex¬ 
erted upon B C. 

Correct ideas of the comparative press¬ 
ure exerted upon timbers according to 
their position, will be readily formed by 
drawing various designs of framing, and 


POCKET COMPANION. 


6'5 


estimating the several strains in accord¬ 
ance with these principles. 

The length of the timber used as posts, 
or struts, does not alter the amount of 
pressure; hut it must be observed, that 
long timbers are not so capable of resis¬ 
tance as short ones. 

u A beam laid horrizontally, supported 
at each end and uniformly loaded, is sub¬ 
ject to the greatest strain at the middle of 
its length. The amount of pressure at 
that point is equal to half the whole load 
sustained. The greatest strain coming 
upon the middle of such a beam, mortises, 
large knots and other defects, should be 
kept as far as possible from that point; 
and, in jesting a load upon a beam, as a 
partition upon a floor beam, the weight 
should be so adjusted that it will bear at 
or near the ends, by means of braces, trim¬ 
mers &c. 

The Resistance of Timber. 

“ When the stress that a given load ex¬ 
erts in any particular direction, has been 
ascertained, before the proper size of the 
timber can be determined for the resist¬ 
ance of that pressure, the strength of the 


THE CARPENTER’S ANI) JOINER’S 


GG 


kind of timber to be used must be known. 

The following rules for calculating the 
resistance of timber, are based upon the 
supposition that the timber used be straight 
grained , seasoned , and free from large knots 
splits, decay &c. The strength of a piece 
of timber, is to be considered in accordance 
with the direction in which the strain is ap¬ 
plied upon it. When it is compressed in the 
direction of its length , its strength is termed 
the resistance to compression. 

When the force tends to pull it asunder in 
the direction of its length, it is termed the 
resistance to tension. And when strained 
by a force tending to break it crosswise, its 
strength is termed the resistance to cross 
strains. 

Resistance to Compression. 

“ When the height of a piece of timber 
exceeds about ten times its diameter if 
round, or ten times its thickness if rectan¬ 
gular, it will bend before crushing. 

The following (Case 1.) refers to such 
posts as would be crushed if overloaded, 
and the other two ‘ cases’ to such as would 
bend before crushing. In estimating the 
strength of timber for this kind of resistance 
it is provided in the following rules that the 


POCKET COMPANION. 


*7 


pressure be exactly in a line with the axis 
of the post. 

41 Case I.— To find the area of a post 
that will safely bear a given weight, when 
the height of the post is less than 10 times 
its least thickness. 

Buie .—Divide the given weight in pounds 
by 1000 for pine, or 1400 for oak, and the 
quotient will be the least area of the post 
in inches. This rule requires that the area 
of the abutting surface be equal to the re¬ 
sult ; should there be, therefore, a tenon on 
the end of the post, this quotient will be too 
small. 

Example .— What should be the least area 
of a pine post that will safely sustain a 
load of 48,000 pounds? 

48,000, divided by 1,000, gives 48—the re¬ 
quired area in inches. Such a post may be 
6x8 inches, and will bear to be of any 
length within 10 times 6 inches, its least 
thickness. 

Case II.— To find the area of a rectangu¬ 
lar post that will safely bear a given load, 
when its height is 10 times its least thick¬ 
ness or more. 

Buie .—Multiply the given weight or press¬ 
ure in pounds, by the square of the length 


68 THE CARPENTER’S AND JOINER’S 


in feet; and multiply this product by the 
decimal, *0015 for oak, *0021 for pitch pine, 
and *0016 for white pine; then divide this 
product by the breadth in inches, and the 
cube root of the quotient will be the thick¬ 
ness in inches. 

Example .— What should be the thickness 
of a pine post, 8 feet high, and 8 inches 
wide, in order to support a weight of 26,880 
pounds? The square of the length is 64 
feet; this, multiplied by the weight in 
pounds, gives 1,730,320; this product, 
multiplied by the decimal, -0016 gives 
2768‘512; and this again, divided by the 
breadth in inches, gives 346-064; by refer¬ 
ence to a table of cube-roots, the cube-root 
of this number will be 7, rejecting decimals. 
Therefore, the post should be, at least 7x8 
inches square. 

Case III.— To find the area of a round , 
or. cylindrical,, post, that will safely bear a 
given weight, when its height is ten times 
its least diameter, or more. . 

Buie .— Multiply the ’given weight or 
pressure in pounds by 1*7, and the product 
by '0015 for oak, -0021 for pitch pine, and 
*0016 for white pine; then multiply the 
square root of this product by the height in 
feet, and the square root of the last product 


POCKET COMPANION. 


69 


will be the required diameter in inches. 

Example .— What should be the diameter 
of a cylindrical oak post, 8 feet high, in or¬ 
der to support a weight of 26,880 pounds? 

This weight in pounds, multiplied by T7 
gives 45,696; and this, multiplied by the 
decimal, *0015 gives 68,544; the square root 
of this product is 8*28, nearly; which, if 
multiplied by 8—the height of the post— 
gives 66*24; the square root of this number 
is 8*14, nearly; therefore, 8*14 inches is the 
diameter required. 

Experiments have shown that the press¬ 
ure should never be more than 1000 pounds 
per square inch on a joint in yellow pine, 
when the end of the grain of one piece is 
pressed against the side grain of the other. 

Resistance to Tension. 

“ A bar of oak an inch square, pulled in 
the direction of its length, has been torn 

asunder by a force of.. 11,500 pounds. 

White pine *. 11,000 “ - 

Pitch pine. . 10,000 u 

Therefore, when the strain is applied in 
a line with the axis of the piece, the follow¬ 
ing rules must be observed: 





70 THE CARPENTER’S AND JOINER’S 


To find the area of a piece of timber to 
resist a given strain in the direction of its 
length. 

Rule .— Divide the given weight to be 
sustained, by the weight that will tear 
asunder a bar an inch square of the same 
kind of wood as above, and the quotient 
will be the area in inches that will just sus¬ 
tain the given weight; but, as a beam or 
post should never be strained or loaded to 
more than one fourth of its breaking weight, 
we must multiply the last area by 4. 

Example .—What should be the area of a 
stick of pitch pine timber, which is required 
to sustain safely a constant load of sixty 
thousand pounds ? This number, divided 
10,000, gives 6, and this, multiplied by 4, 
gives 24 inches-the area, or a stick 4x6. 

Resistance to Cross Strains. 

To find the Scantling of a piece of timber 
to safely sustain a given weight, when 
such piece is supported at the ends in a 
horrizontal position. 

Case I.— When the breadth is given. 

Rule .—Multiply the square of the length 
in feet, by the weight in pounds, and this 


POCKET COMPANION. 


71 


product by the decimal, *009 for oak, *011 
for white pine, and’016 for pitch pine; di¬ 
vide the product by the breadth in inches, 
and the cube root of the quotient will be 
the depth required in inches. 

Example .— What should be the depth of 
a beam of white pine, in order to support 
900 lbs., having a bearing of 24 feet, and a 
depth of 6 inches? 

The square of 24 is 576, and this, multi¬ 
plied by 900, gives 518,400; and this again, 
by *011 gives 5702* 400; this, divided by 6. 
gives 950’400; the cube root of which is 
9*83 inches, the depth required. 

Case II,— When the depth is given. 

Buie .— Multiply the square of the length 
in feet by the weight in pounds, and mul¬ 
tiply this product by the decimal, • 009 for 
oak, - 011 for white pine, and *016 for pitch 
pine; divide the last product by the cube of 
the depth in inches, and the quotient will be 
the breadth in inches required. 

Example .— What should be the breadth of 
a beam of oak, having a bearing of 16 feet 
and a depth of 12 inches, in order to support 
a weight of 4000 lbs. ? The square of 16 is 
256, which, multiplied by 4000, gives 
1,024,000; this, multiplied by *009 gives 
9216; and this again divided by 1728, — the 


72 


THE CARPENTER’S AND JOINER'S 


cube of 12,—gives 5% inches,—the breadth 
required. 

Note. — The greater the depth of a beam 
in proportion to the thickness. The great¬ 
er the strength, provided it be thoroughly 
stayed so as to prevent it from falling over 
and breaking sideways. A beam 3 x 16 in¬ 
ches square would bear twice as much as a 
square beam of the same section, which 
shows hov r important it is to make beams 
deep and thin. 

To cur a Bake Crown Moulding so 
as to miter with a Plumb Eave. 



A A 

FIG. 14 


Make the plumb cuts down the sides of 
the box as shown at A, A, fig. 14, and 
cut the miter across the top as usual, but, 
remember, a square miter will not do. To 
get the bevel for the miter: proceed accord¬ 
ing to the method of getting the side bevel 
for Jack Rafters of same pitch roof. (See 
page 49, for side bevel of Jacks.) This rule 
applies to any pitch, by taking the pitch 
line of one foot run, and taking this with 





POCKET COMPANION. 


73 


12 to obtain the miter,—using the long bevel. 
The same bevel will also do for mitering 
Rake Facia and Frieze, or any raking 
member to a level one standing plumb. 


TO FIND THE LENGTH OF ROD, OK DIST¬ 
ANCE between Tram Points 
for striking Segments. 


D 



The usual method among workmen is, 
after determining the three points A, B, 
and D, fig. 15 to draw lines at right angles 
to, and from the center of each of the Chords 
of A D, and D B, taking the point of inter¬ 
section of these lines for the radius. This 
is very well for small segments, but suppose 
you have a very large segment to strike, 
you will find it difficult to obtain the radius 
by lines , while with figures it becomes an 
easy task. 




74 THE CARPENTER’S AND JOINER’S 


Suppose the Chord AD, to measure 25 in¬ 
ches, and the Versed sine CD, 8 inches; 
square the 25 inches ( which is done by 
multiplying this number by itself) and 
then divide the product by the rise in 
inches, as 8. 


Demonstration . 

Let A D =25 
25 

C D =8)625 being the square of 25- 

78^ inches, which would be 
the diameter of the circle, of which the arc, 
A D B, is a portion. Then, the Baclius , or 
distance between the Tram-points wonld be 
one-half the diameter obtained. 

Where feet and inches, or fractions of an 
inch occur in the measurements obtained at 
A D, or C D, both terms must be reduced 
to the lowest denomination contained in 
either of the measurements; thus, if A D, 
equals 2534 inches, and C D, equal 6%. in¬ 
ches, then each term must be reduced to 
eighths before proceeding any farther; 
then, when the last result, *or diameter, is 
obtained, it can be divided by 8, in order 
to bring it back to inches. The same re¬ 
sult can be obtained more readily by re¬ 
ducing the fraction, to decimals. 




POCKET COMPANION. 


75 


Explanation of Arithmetical Signs 

GENERALLY USED IN MATHEMATICAL 

Calculations, to which particular 

ATTENTION SHOULD BE GIVEN. 

= Sign of equality, or equal to. 

+ u Addition, plus, or more. 

— “ Subtraction, minus, or less. 

x “ Multiplication. 

-p 4 ‘ Division. 

: Is to. ) 

:: So is. V Signs of Proportion. 

: To. ) 

y' Square root; when placed before a 
number, the square root is to be extracted, 
as, .y/64=8. 

Cybe root; and signifies that the 
cube, or third root is to be extracted, as 
3^64=4. 

A number is said to be squared, when it 
is multiplied by itself. To cube a number, 
is, to multiply it three times by itself, as, 
the cube of 4 is 4 x 4 x 4=64. 

O Degrees, ' minutes, " seconds. 

In Duodecimals, ' denotes primes, or 
twelfths, "seconds, or twelfths of primes, 

thirds, or twelfths of seconds; thus, the 
term, 4 6' 3" reads, 4 ft, 6 in. and 3 twlfths. 

* Decimal point, as, .5= five tenths 
*05= five hundredths, 2*8= 2 and 8 lOths. ’ 


The Circle; its relative proportions, Areas &c. 


76 


the carpenter’s and joiner's 


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POCKET COMPANION. 


7 < 


/ 


The Slide Rule. 

The Slide Rule is a two-foot rule having 
only one joint, and is constructed as follows: 

The line A is laid down twice along the 
top of the rule; the line 0 once in the same 
space at the bottom of the rule. Between 
them is a groove for the reception of the 
slide B C, which is merely a copy of the 
line A. There is a mutual relation between 
the lines which will be readily understood 
by attending to the remarks that follows. 

When the slide B C, is laid evenly in the 
groove, that is, when the beginning of A 
coincides with the beginning of B, the 
numbers on A are the same as the numbers 
on B; when the slide is in any other position, 
the numbers on A are proportional to the 
numbers on B. The same is the case with 
the D line; when the slide lies evenly in 
the groove, the numbers on C, are the 
squares of those on D ; and consequently, 
the numbers on D are the square roots of 
those on C. When the slide is in any other 
position, the numbers on C ar oproportional 
to the squares of those on D. 

The values of the numbers on those lines 
are governed by the value set upon the 
first figure, and, being reckoned decimally 


78 THE CARPENTER’S AND JOINER’S 


\ 


advance ten-fold. Thus, suppose 1 at the 
joint to be 1, then 1 at the middle of the 
rule is 10, and 10 at the end of the rule is 
100; again, suppose 1 at the joint is 10 ? 
then 1 at the middle is 100, and 10 at the 
end is 1000; the intermediate divisions on 
which, complete the whole system of its 
notation. 

Examples for practice. 

To multiply numbers by the Rule. 

Buie .—Set 1 on B opposite the multiplier 
on A, and against the number to be multi¬ 
plied on B, is the product on A. 

Example. — 234 x 23 4 are how many ? 

Set 1 on B against 234 on A, and against 
2% on B is 634 011 A. 

To divide numbers by the Kule. 

Buie. —Set the divisor on B to 1 on A. and 
against the number to be divided on B, is 
the quotient on A. 

Example. — Divide 9 by 234- 
Set 234 on B against 1 on A, and against 
9 on B is 4 (the quotient) on A. 


POCKET COMPANION. 


71 ) 


Direct Proportion : By the Rule. 

Rule .—Set the first term on B to the 
second on A, and against the third on B is 
the fourth on A. 

Example .— 3 is to 5, as 6 is to what num¬ 
ber? 

Against 5 on A set 3 on B ; against 6 on 
B is 10 (the answer) on A. 

Inverse Proportion. 

Rule .— Invert the Slide, and then the 
operation is the same as Direct proportion. 

Example .— If 6 men can build a house in 
12 days, how many could do the same work 
in 8 days ? 

Invert the Slide, and set 6 of 0 against 
12 on A; against 8 on C is 9 on A. 

Mensuration of Surfaces. 

Rule .— When the length is given in feet, 
and the breadth in inches: Set. the breadth 
on B against 12 on A, and against the 
length on A is the content in square feet on 

B. 

Note.— If the dimensions are all inches, 
set the breadth on B against 144 on A, and 
against the length on A is the content in 
square feet on B. 


80 


TIIK CARPENTER'S AND JOINER’S 


Example .— How many feet in a board 14 
inches wide and 14 feet long ? 

Set 14 of B against 12 on A, and against 
14 on A is 16% on B. 

Circles. 

Set *7854 of C against 10 on D; then C is 
a table of areas, and D a table of diameters. 
If 7 of B is set against 22 on A, then, A is a 
table of circumferences, opposite the diame¬ 
ters on B. 

Solid Content of Round Tim her. 

Buie .— Measure the log around the mid¬ 
dle, and call one-quarter of this the girt. 
Then set 12 of D to the length in feet on C, 
and against the girt in inches on D, is 
the solid content in feet on C. 

The lines, M, and E, on the opposite side 
of the Rule are useful, either for laying 
out Octagons, or getting the length of 
Braces, fully explained on pages 44, 45. 

With the Slide Rule, a great variety of 
operations may be performed, and, with a 
little practice, will soon be found to be of 
greater value than is generally supposed 
among Mechanics. Those Rules on which 
the D line commences with 1 are the best 
adapted to general use. 


POCKET COMPANION. 


*1 


An Abridged, but Comprehensive Sys¬ 
tem of Arithmetical Rules. 

The following Rules and Memorandas 
are only intended to assist the memory of 
those who already understand the funda¬ 
mental rules of common Arithmetic, and 
not for the new beginner; therefore all ex¬ 
amples and demonstrations have been omit¬ 
ted. 

Decimal Fractions. 

Decimal fractions originate from divid¬ 
ing 1 into 10 equal parts, called tenths ; each 
of these tenths is divided into 10 equal parts 
called hundredths ; each of these hundredths 
is divided into 10 equal parts, called thous¬ 
andths; and so on. 

In writing Decimals, the denominator is 
omitted, and a point (.) called the Decimal 
Point ,is placed before the numerator sepa¬ 
rating it from the whole number ( when 
the fraction is compound ). Thus five and 
six tenths is written 5.6—and seventy five 
hundredths is written .75,&c. 

Cyphers added to the Bight hand of a 
decimal do not increase its value nor affect 
it in any way; but each cypher prefixed to 
a decimal fraction decreases its value ten¬ 
fold. 


82 


THE CARPENTER’S AND JOINER’S 


Addition and Subtraction of Decimals. 

Buie .— Place the numbers under each 
other according to their value; that is, 
tenths under tenths, hundredths under 
hundredths, &c., and proceed the same as 
in common addition and subtraction. 

Multiplication of Decimals. 

Write the numbers under each other as 
if they were whole numbers , and then pro¬ 
ceed as in multiplication of simple numbers. 
Then point off in the product as many pla¬ 
ces for decimals as there are decimals in 
both factors. It may be sometimes neces¬ 
sary to prefix cyphers to the decimal part 
of the product, in order to have the num¬ 
ber of decimal places equal to those of the 
two factors. 

Division of Decimals. 

Buie .—Proceed as in simple numbers, and 
point off in the quotient as many decimals 
' as the number of decimal places in the divi¬ 
dend exceed those in the divisor, supplying 
any deficiency by prefixing cyphers to the 
left of the quotient. 


POCKET COMPANION. 


83 


To Reduce a Vulgar Fraction 
To a Decimal. 

Buie .—Add to the numerator of the given 
fraction any number of cyphers at pleasure 
and divide the numerator by the denomin¬ 
ator; then the quotient will he the decimal 
of equivalent value. 

To find the Fractional Value 
Of a Decimal. 

Buie .—Multiply the given decimal by the 
various fractional denominations of the 
integer, or whole number, cutting off from 
the right hand of each product, for deci¬ 
mals, a number of figures equal to the given 
number of decimals, and thus proceed until 
the lowest degree is obtained. 


DUODECIMALS, 

Duodecimals are denominate numbers, 
the denominations of which increase accord, 
ing to the scale of 12. 

The foot is the unit of measure. Each 
foot is divided into 12 equal parts called 
primes, each prime into 12 equal parts 
called seconds, and so on with the lower 
denominations, 1 being equal to 12 of the 
next lower denomination. 



84 THE CARPENTER’S AND JOINER’S 


Multiplication of Duodecimals. 
Rule .— Place units of the same value un¬ 
der each other, that is, feet under feet, 
inches under inches, &c. Begin at the right 
hand, multiplying each term of the multi¬ 
plicand by the lowest denomination of the 
multiplier, reducing each product to the 
next higher term. Then take the next high¬ 
er denomination of the multiplier and 
multiply each term of the multiplicand by 
that also, and so on, until the process will 
be completed; then add together the sever¬ 
al products, carrying the excess of 12s from 
the right hand to the next higher term at 
the left. 

Feetf xfeet gives feet. 

“ x inches “ inches. 

“ x twelfths “ twelfths. 

Inches x inches “ “ or seconds. 

“ x seconds “ thirds. 

Seconds x “ “ fourths &c., &c. 

Division of Duodecimals. 

Rule .—First find out how many times 
the highest term of the divisor is contained 
in the highest term of the dividend; then 
multiply the whole divisor by this one term 
of the quotient, and subtract the product 
from the dividend. To the result bring 
down the next term of the dividend, and 
divide as before. 


POCKET COMPANION. 


85 


SIMPLE PROPORTION. 

A Simple Proportion is an expression of 
equality between two simple and equal 
ratios. Thus, 3 : 4 :: 6 : 8 form a pro¬ 
portion. The terms are read, 3 is to 4 as 
6 is to 8. The 1st and 4th terms are called 
the Extremes; the 2d and 3d, the Means. 
Thus, in the proportion given above, 3 and 
8 are the extremes , and 4 and 6 the means. 

In any Proportion, the product of the 
means is equal to the product of the ex¬ 
tremes. Either extreme is equal to the pro¬ 
duct of the means divided by the other ex¬ 
treme. Either mean is equal to the product 
of the extremes divided by the other mean. 

The fourth term of a proportion is equal 
to the third term divided by the ratio of 
the first to the second. 

Buie. —I. Call the required answer, 
and write it in the 4th place, and the term 
having the same unit value, (i.e. of like kind) 
in the 3d place: 

II. Then analyze the question, and see 
whether the 4th term is greater or less than 
the 30; when greater, write the least of the 
remaining terms in the 1st place, and when 
less, write the greater there, and the re¬ 
maining term in the 2d place. 


86 THE CARPENTER’S AND JOINER’S 


III. Then multiply the 2nd and 3rd 
terms together, and divide the product by 
the 1st. 

Therefore, according to the preceding 
Buies , we have the following: 

Formulas. 

Quantity : Quantity :: Cost : Cost. 

Labor : Labor :: Time : Time. 
Labor : Labor :: Work done : Work done. 

If A : B :: C : D, then, 
A*xDf—BxC. And BJ-*-A=D-^C. 

AxD-fB=C. And AxD-*-C:=B. 

BxCh-D=A. AndBxC-^A=D. 

Thus, the terms BxC-s-A=D; means 
that B multiplied by C and their product 
divided by A equals D. 


FRACTIONS. 

A fraction is composed of two terms, 
called the Numerator and Denominator. 

Thus, %; 3 being the numerator, and 4 
the denominator. 

If both terms of a fraction be multiplied 
by the same number, its value wil not be 
changed. 

^Multiplied by. tEqual to. ^Divided by. 




POCKET C(>MFANION. 


87 


If both terms of a fraction be divided by 
the same number, the value of the fraction 
will not be changed. 

To reduce a mixed number to an equiva¬ 
lent improper fraction. 

Buie .—Multiply the whole number by the 
denominator; to the product add the nume¬ 
rator, and place the sum over the given 
'denominator. 

To reduce an improper fraction to an 
equivalent whole or mixed number. 

Buie .—Divide the numerator by the de¬ 
nominator, and the quotient will be the 
equivalent whole or mixed number. 

To reduce a fraction to its lowest terms. 

Buie .—Divide the numerator and denom¬ 
inator, successively, by all their common 
factors: Or, 

Divide the numerator and denominator 
by their greatest common divisor. 

To reduce a compound fraction to a sim¬ 
ple fraction. 

j Rule. —I. If there are mixed numbers, 
reduce them to improper fractions. 

II. When there are common factors in 
the numerators and denominators, cancel 
them. Then, multiply the numerators to¬ 
gether for a new numerator, and the de¬ 
nominators together for a new denominator 




88 THE CARPENTER’S AND JOINER’S 


Curious and Useful Contractions. 

To multiply any number of two figures 
by 11: Write the sum of the figures between 
them. Thus,—34x11 =374. 

Here we simply say, 3 and 4 are 7, and 
then place the 7 between the 3 and the 4. 

N. B. When the sum of the figures is 
more than 9, increase the left-hand figure 
by the 1 to carry. 

To square any number of 9s immediately 
without multiplying: Set down as many 
9s less one (beginning at the left,) as there 
are 9s in the given number, an 8, as many 
Os as you do 9s, and a 1. 

To square any number ending in 5: 

Omit the 5, and multiply the number as 
it will then stand, by the next higher num¬ 
ber, and annex 25 to the product. 

Thus. 35x35=1225: We simply say, 
3 times 4 are 12, and annex the 25. 

To square any number containing 34 , as 
534 &c: Multiply the whole number by 
the next higher whole number, and annex 
M to the product. Thus, 534 x 634 = 30 * 4 . 
We simply say, ‘5 times 6 are 30’ to which 
we add 34- 


POCKET COMPANION. 


89 


To multiply any two like numbers to¬ 
gether when the sum of the fractions is one. 

Buie .—Multiply the whole number by the 
next higher whole number, and annex the 
product of the fractions. Thus, 

4%x4h£ =20 and 3 16ths. 

Rapid process of multiplying all mixed 
numbers.—This is a general rule. 

1st. Multiply the whole numbers together. 
2 nd. Multiply the upper whole number by 
the lower fraction. 

3rd. Multiply lower whole number by the 
upper fraction. 

4th. Multiply the fractions together. 

5th. Add these four products together. 

To multiply any tw o numbers together 
when the sum of the unit figures is ten , and 
the other figures of both factors are alike. 
Thus.—47x43 =2021. 

We simply say 4 times 5 are 20, and 
3 times 7 are 21, and set down each product 
successively. 

By this, it will be observed, that we must 
always increase the upper tens figure 1 be¬ 
fore multiplying. 

To multiply any number by 125 add three 
ciphers, and divide by 8. 


90 the carpenter’s and joiner’s 


To multiply any number by 16% add two 
ciphers, and divide by 6.' 

To multiply any number by 33% add two 
ciphers and divide by 3. 

To multiply any number by 166% add 
three ciphers, and divide by 6. 

To multiply any number by 333% add 
three ciphers, and divide by 3. 

To multiply any number by 6% add two 
ciphers, and divide by 15; or add one cipher 
and multiply by %. 

To multiply any number by 66% add three 
ciphers, and divide by 15; or add two ci¬ 
phers and multiply by %. 

To multiply any number by 8% add two 
ciphers, and divide by 12. 

To multiply any number by 83% add three 
ciphers, and divide by 12, 

To multiply any number by 6% add two 
ciphers, and divide by 16 or its factors—4 x4. 

To multiply any number by 62% add 
three ciphers, and divide by 16. 

To multiply any number by 18% add 
two ciphers, and multiply by 3, and divide - 
by 16, or its factors, 4x4. 

To multiply by 37% add two ciphers and ‘ 
multiply by 3, and divide by 8. 


To multiply any number by 87*4 add two 
ciphers, divide by 8. and subtract the quo¬ 
tient. 

To multiply any number by 75 add two 
ciphers, divide by4, and subtract the quo¬ 
tient. 

To multiply by 09 add two ciphers, a n d 
subtract the given number from the result. 

To multiply by any number of 9's. 

Rule. —Annex as many ciphers to the 
multiplicand as there are 9’s in the multipli¬ 
er. and from this number subtract the num¬ 
ber to be multiplied, and the remainder is 
the product required. 

Ojtoss Multiplication. 

Cross multiplication is a method of mul¬ 
tiplying large numbers in a single line. 

Here we multiply 5x4. set down 24 
the 0 and carry 2: next, 5x2 and 3x4 35 

which, with the 2 to carry, gives 24. - 

set down 4 and carry 2; last. 3x2 and 840 
add the 2 to carry, making 8. 

Here wo first multiply 7x4 234 

ibid., 7x3 and 6x4. 3rd.. 7x2. 5x4. 567 

and 6x3. 4th., 6x2 and 5x3. - 

5th and last, 5x2, carrying the ex- 132678 
cess of tens at each multiplication. 




92 THE carpenter’s and joiner’s 


MISCELLANEOUS MEMOKANDAS 

PERTAINING TO 

MEASUKES, CAPACITIES &C. 

DISTILLED WATER. 

One gab = 10 lbs. = 277*75 cubic inches. 

WINE MEASURE. 

One gallon = 231. cubic inches. 
One quart = 57*75 “ “ 

One pint = 28*875 “ “ 

DRY MEASURE. 

One bushel = 2150*42 cubic inches. 
One gallon = 268*80 “ “ 

One quart = 67*20 “ “ 

One pint = 33*60 “ “ 

A heaping bushel of ears of corn, or roots> 
contains about 2,818 cubic inches.. 

BINS FOR GRAIN. 

Knowing the number of bushels, to find 
the number of cubic feet which the bin 
must contain. 

Increase the number of bushels one-fourth 
itself, and the result will show the number 
of cubic feet. 

To find the number of bushels which a 
bin of given dimensions will hold. 

Diminish the number of cubic feet in the 
bin by one-fifth, and the result will be con¬ 
tent in bushels. 


POCKET COMPANION. 


93 


To find the number of cubic feet in a bin, 
or box. Multiply the length by the width, 
and that product by the depth. 

Two cubic feet of good, sound, dry corn 
in the ear, will make a bushel of shelled 
corn. 


Kelative Weights of Metals. 


Cubic inches multiplied by: 
*263 = pounds of cast iron. 


•281 = 
•283 = 
•3225= 
•3037= 
•26 = 
•4103= 
•2636= 
•4908= 


wrought iron. 

steel. 

copper. 

brass. 

zinc. 

lead. 

tin. 

mercury. 


Cylindrical inches multiplied by: 


•2065= 

i< 

cast iron. 

•2168= 

a 

wrought iron, 

•2223= 

tt 

steel. 

•2333= 

<t 

copper. 

•2385= 

u 

brass. 

•2042= 

u 

zinc. 

•3223= 

•t 

lead. 

•207 = 

a 

tin. 

•3854= 

a 

mercury. 


94 


THE CARPENTER'S AND JOINER’S 


Weight of Water at its 
Common Temperature. 


1 cubic inch 

= 

•03617 lbs. 

12 

“ inches 

= 

•434 

1 

“ foot 

= 

62-5 “ 

1 

u u 

= 

7-81 gallons. 

1-6 

“ feet 

= 

100 lbs. 

32 

u ii 

= 

1 ton. 

1 cylindrical inch 

= 

•02842 It s. 

12 

“ inches 

= 

•341 “ 

1 

“ foot 

= 

49-1 

1 

it u 

= 

6*136 gallons. 

2-036 

“ feet 


100 lbs. 

40-73 

U l< 

= 

1 ton. 

12*5 gallons 

= 

100 lbs. 

150 

a 

= 

1 ton. 


Bricklayer’s Memoranda. 

A load of mortar measures 1 cubic yard, 
or 27 cubic feet; requires 1 cubic yard of 
sand, and 9 bushels of lime; and will fill 30 
hods. 

A bricklayer’s hod measuring 1 foot 4 
inches x 9inches x 9 inches; equals 1.296 
cubic inches in capacity; and contains 20 
bricks. 

A single load of sand and other materials 
equals 1 cubic yard, or 27 cubic feet. And a 
double load equals twice that quantity. 


POCKET (UMPAMON, 


9f) 


ELEMENTS OE MACHINERY. 

There are six simple machines, which are 
called Mechanical powers. They are the 
Lever , the Pulley , the Wheel and Axle, the 
Inclined Plane, the Wedge, and the Screw. 

To understand the nature of a machine, 
four things must be considered. 

1st The power or force which acts. 

This consists in the efforts of men or 
horses, of weights, springs, steam, &c. 

2d. The resistance which is to be over¬ 
come by the power. This generally is a 
weight to be moved. 

3d. The center of motion, called a .fulcrum 
or prop. 

4th. The respective velocities of the pow¬ 
er and resistance. 

A machine is said to be in equilibrium 
when the resistance exactly balances the 
power. 


THE LEVER. 

An equilibrium is produced in any lever, 
when the weight, multiplied by the dist¬ 
ance from the fulcrum, is equal to the pow¬ 
er multiplied by the distance from fulcrum 
That is, 

The weight is to the power, as the distance 


THE CARPENTER’S AND JOINER’S 


96 


from the power to the fulcrum, is to the 
distance from the weight to the fulcrum. 

Hence, if A=weight; B=power; 
C=distance from power to fulcrum; 
and D=distance from weight to fulcrum; 

Then, A: B :: C: D 

THE PULLEY. 

Pulleys are divided into two kinds, fixed pul¬ 
leys and movable pulleys. When the pulley 
is fixed, it does not increase the power 
which is applied to raise the weight, hut 
merely changes the directi on in whi eh i t acts 
A movable pulley gives a mechanical ad¬ 
vantage. Thus a weight attached to a 
single movable pulley, having a cord passed 
around it, one end being fastened to a hook, 
and the hand or power at the other end, 
then, the hook will sustain one-half the 
weight, the hand sustains the other half. 

It is plain, that in the movable pulley, 
all the parts of the cord will be equally 
stretched; and hence, each cord running 
from pulley to pulley, will bear an equal 
part of the weight; consequently, 

Buie .—The power will always be equal 
to the weight divided by the number of 
cords which read) from pulley to pulley. 


POCKET COMPANION. 


97 


THE WINDLASS; OR WHEEL AND AXLE. 

Iii order to balance the weight, we have 
the following: 

Rule .—The power is to the weight, as the 
radius of the axle, is to the length of the 
crank, or radius of the wheel. 

THE INCLINED PLANE. 

Rule .—The power is to the weight, as the 
height of the plane is to its length. 

THE WEDGE. 

Rule .—Half the thickness of the head of 
the wedge, is to the length of one of its 
sides, as the power which acts against its 
head is to the effect produced at its side. 

THE SCREW. 

Rule .— As the distance between the 
threads of a screw, is to the circumference 
of the circle described by the power, so is 
the power employed to the weight raised. 

LAWS OF FALLING BODIES. 

A dense body falling vertically through 
a vacuum , falls 16 and one-twelfth feet dur¬ 
ing the first second, 48 and one-quarter feet 
during the second. 80 and five-twelfths feet 
the third second, and so on : the speed in¬ 
creasing at the rate of 32 and one-sixth feet 
for each second of time. 


98 TH*£ CARPENTER’S AND JOINER*S 


GENERAL RULES FOR CALCULAT¬ 
ING THE SPEED AND DIAMETER 
OF PULLEYS. 

PROBLEM 1ST. 

The diameter of a pulley, and the speed 
it is required to run, with the speed of the 
shaft that is to drive it, given, to find the 
si/e of the driving pulley. 

Rule .— Multiply the diameter of the pul¬ 
ley you have, by the speed that it is re¬ 
quired to run, and divide this product by 
the speed of the shaft that is to drive it, 
the quotient will be the diameter required. 

PROBLEM 2nd. 

The diameter of two pulleys given, and 
the speed of one to find the speed of the 
other. 

Rule.r- Multiply the diameter of the pul¬ 
ley whose speed is given, by it.- number of 
revolutions per minute, and divide the pro¬ 
duct by the diameter of the pulley whose 
speed is not given, and the quotient will be 
the number of revolutions per minute. 

Or, divide the greater diameter by the 
lesser diameter, and the quotient will be 
the number of revolutions the lesser will 
make, for one of the greater. 


POCKET COMPANION. 


99 


To Soften Putty and remove Paint. 

To destroy paint on old doors &c„, and 
to soften putty in old sash, so that the 
glass may he easily removed:— Take one 
part American Pearlash, and three parts of 
quick stone lime; slack the lime in water, 
than add the pearlash, making the mixture 
about the consistancy of paint. 

Apply on both sides of the glass, or over 
the whole surface of old paint, letting it re¬ 
main on 12 or 14 hours. 

Quantity of Water in Pipes. 

A pipe of one inch in diameter, and one 
yard in length, contains 28^ cubic inches, 
or very nearly a pound or water. 

Hence, the following practical rule is 
generally used to find the quantity of water 
in a pipe of any diameter. 

Rule .—Square the diameter of the pipe in 
inches, the product will be the weight of 
water in pounds per yard of the pipe’s 
length; shift the decimal point one place 
to the left, and the result is the quantity 
of water in a pipe of anj^ given diameter 
per yard in gallons. 

The content of any cylindrical vessel can 
be found in like manner. 


100 THE CARPENTER’S AND JOINER’S 


Right and Left Hand Doors. 
When standing at the opposite side from 
which the door is hung:— 

If it opens away from you to the right, it 
is said to be a right-howl door. If it opens 
away from you to the left, it is said to be a 
left-hand door. . 

LOOSE JOIKT DOOR BUTTS. 

TO tell either hand at a glance. 
Lay down the butts so that the knuckle- 
joint will be toward the right haud, thus: 

A 



Butts closed. B 

Then, if they are open at the top, as at A 
they are right hand ; if open at the bottom, 
as at B, they are left hand. 

Doors are often affected by an unequal 
temperature of adjoining rooms, causing 
the stiles to spring, or warp; this may be 
remedied by using three hanging butts. 

All doors 7 ft. or more in height, should 
be hung with three butts. 















POCKET COMPANION. 


101 


Table of measures. 

A box 18x18 inches square, and 26% in. 
ches deep, will contain a barrel; true con¬ 
tent. 8467*20 cubic inches. 

A hoop 18% inches in diameter, and 8 in¬ 
ches deep will contain one bushel. 

A box 15%xl5% inches square, and 9% 
inches deep, will contain one bushel. 

A box 10x12 inches square, and 9 inches 
deep, will contain half a bushel. 

A box 8x8 inches square, and 8 inches 
deep, will contain one peck. 

A box 8x8 inches square, and 4% inches 
deep, will contain one gallon. 

A box 4x8 inches square, and 4% inches 
deep, will contain half a gallon. 

A box 4x4 inches square, and 4% inches 
deep, will contain one quart. 

A box 4x4 inches square, and 2% inches 
deep, will contain one pint. 

CUBIC, OR SOLID MEASURE. 

1728 Cubic Inches.1 Cubic Foot. 

27 Cubic Feet.1 Cubic Yard. 

128 Cnbic Feet.1 Cord. 

SQUARE MEASURE. 

144 Square Inches.1 Square Foot. 

9 Square Feet.1 Square Yard. 

100 Square Feet ... one Sq. of Flooriug &c. 







102 THE CARPENTER’S AND JOINER’S 


Tints to Express Building Materi¬ 
als :—As used by architects in draughting. 

Materials. Color. 

Brickwork in plans 
and sections,...Crimson lake. 

Brickwork in eleva¬ 
tions, .Crimsou lake mixed 

with burnt sienna 
or Venetian red. 

The lighter woods, 
such as pine.Raw sieana. 

Oak or ash.Vandyke brown. 

Granite.Pale Indian ink. 

Stone generally.Yellow ochre, or pale 

sepia. • 

Concrete works.Sepia with darker 

markings. 

Wrought iron.Indigo. 

Cast iron.Payne’s grey or neut¬ 

ral tint. 

Steel.Pale indigo tinged 

with lake. 

Brass.Gamboge or Roman 

ochre. 

Lead.Pale Indian ink tinged 

with indigo. 

Clay or earth.Burnt umber. 

Slate.Indigo and lake. 















































' 























